
Nearest Neighbor Matching: We start with the simplest method where for estimating the exposure vector of each [mock] engager we choose their nearest neighbor match from the contributors' group. However, this might not be a good estimate as this might pick up the nearest contributor in which the user would not talk about the same topics that the engager does (Fig.REF ). We are mostly looking for cases in which the engager and the contributor both talked about the same topics, while the contributor might talk about other topics as well. In other words, we want to minimize the cases when we have zero value for a topic in the estimated exposure vector while its corresponding activity value is one (Conditional Nearest Neighbor Matching).

Hierarchical Poisson Factorization: Originally proposed by Gopalan et al. [1]}, Wang et al. [2]} utilized this method for movie recommender systems. The method first uses Poisson Factorization (PF) to fit a model to the observed user-movie ratings (i.e. the user-topic activity vector, in our case). Next, it reconstructs the exposure matrix from the PF fit which accounts for unobserved confounders. More formally, lets assume that \(\pi _u \stackrel{\scalebox {.5}{i.i.d}}{\sim } \text{Gamma}(c_1,c_2)\)  captures user preferences and \(\lambda _t \stackrel{\scalebox {.5}{i.i.d}}{\sim } \text{Gamma}(c_3,c_4)\)  captures topic attributes. Then PF assumes the data come from the following process:
\(a_{ut}|\pi _u, \lambda _t \;\; \sim \;\; \text{Poisson}(\pi _u^T\lambda _t) \quad \forall u,t,\) 
the exposure matrix is reconstructed from the PF fit by:
\(\hat{e}_{ut}=\mathbb {E}_{\text{PF}}[\pi _u^T\lambda _t|\mathbf {a}].\) 

